Question: Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{x^2 + 10x}{x^2 + 19x + 90}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 10x}{x^2 + 19x + 90} = \dfrac{(x)(x + 10)}{(x + 9)(x + 10)} $ Notice that the term $(x + 10)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x + 10)$ gives: $p = \dfrac{x}{x + 9}$ Since we divided by $(x + 10)$, $x \neq -10$. $p = \dfrac{x}{x + 9}; \space x \neq -10$